On the construction of quantum fields via measures on path spaces

Rodrigo Vargas Le-Bert

Published 2014-11-29Version 1

We engage the construction problem of quantum fields via Euclidean measures on path spaces. We start by recasting stochastic process theory in algebro-analytic language, revisiting Nelson's reconstruction theorem after Klein and Landau, and developing a version of the theory of cylinder measures and their radonification which is amenable to explicit calculations. Then, we provide a new construction of the $\phi^4$ field in dimensions $1+1$ and $2+1$, based on the "re-radonification" on a space of continuous, distribution-valued paths, of a measure obtained with the aid of Rajput's characterization of Gaussian measures on $L^p$ spaces. Finally, we develop an understanding of renormalization, based on our notion of cylinder measure, which we exploit in two ways. First, we introduce the notion of cylinder perturbation and sketch a proof that the $\phi^4$ field can exist in dimension $3+1$ only in a weak coupling limit. Second, we devise a perturbation series on the cutoff scale and propose an explicit solution of the renormalization problem, by treating the interaction part alone. As an application, we provide a formula for the cylinder measure of a local field which is effectively $\phi^4$ at any given fixed scale, in arbitrary dimension.